∠BED = ∠BAE+∠ABE = 90°+45° = 135° ,
∠DCE = 90°-∠CED = 90°-(1/2)∠BED = 22.5° ,
tan22.5° = tan(45°/2) = (1-cos45°)/sin45° = √2-1 ,
CD = AB = 1 ,
AE = AB*tan∠ABE = AB*tan45° = 1 ,
DE = CD*tan∠DCE = CD*tan22.5° = √2-1 ,
AD = AE+DE = √2 .